On $\mathcal{I}_2$-Convergence and $\mathcal{I}_2^{*}$-Convergence of Double Sequences in Fuzzy Normed Spaces

On $\mathcal{I}_2$-Convergence and $\mathcal{I}_2^{*}$-Convergence of Double Sequences in Fuzzy Normed Spaces

In this paper first, we investigate some properties of $\mathcal{I}_2$-convergence in fuzzy normed spaces. After, we study some relationships between $\mathcal{I}_2$-convergence  and $\mathcal{I}_2^{*}$-convergence of double sequences in fuzzy normed spaces.

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  • [1] Bag, T. and Samanta, S.K., Fixed point theorems in Felbin's type fuzzy normed linear spaces, J. Fuzzy Math. 16(1) (2008), 243-260.
  • [2] Bede, B. and Gal, S.G., Almost periodic fuzzy-number-valued functions, Fuzzy Sets Syst. 147(2004), 385-403.
  • [3] Das, P., Kostyrko, P., Wilczynski, W. and Malik, P., I and I*-convergence of double sequences, Math. Slovaca, 58(5) (2008), 605-620.
  • [4] Das, P. and Malik, P., On extremal I-limit points of double sequences, Tatra Mt. Math. Publ. 40 (2008), 91-102.
  • [5] Diamond, P. and Kloeden, P., Metric Spaces of Fuzzy Sets-Theory and Applications, World Scienti c Publishing, Singapore (1994).
  • [6] Dündar, E. and Altay, B., I2-convergence and I2-Cauchy of double sequences, Acta Mathematica Scientia, 34(2) (2014), 343-353.
  • [7] Dündar, E. and Altay, B., On some properties of I2-convergence and I2-Cauchy of double sequences, Gen. Math. Notes, 7(1) (2011), 1-12.
  • [8] Dündar, E. and Talo,  O., I2-convergence of double sequences of fuzzy numbers, Iranian Journal of Fuzzy Systems, 10(3) (2013), 37-50
  • [9] Dündar, E. and Talo,  O., I2-Cauchy Double Sequences of Fuzzy Numbers, Gen. Math. Notes, 16(2) (2013), 103-114.
  • [10] Fang, J.-X. and Huang, H., On the level convergence of a sequence of fuzzy numbers, Fuzzy Sets Systems, 147 (2004), 417-415.
  • [11] Fast, H., Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.
  • [12] Felbin, C., Finite-dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48(2) (1992), 239-248.
  • [13] Hazarika, B., On ideal convergent sequences in fuzzy normed linear spaces, Afrika Matematika, 25(4) (2013), 987-999.
  • [14] Hazarika, B. and Kumar, V., Fuzzy real valued I-convergent double sequences in fuzzy normed spaces, Journal of Intelligent and Fuzzy Systems, 26 (2014), 2323-2332.
  • [15] Kostyrko, P., Salat, T. and Wilczynski, W., I-convergence, Real Anal. Exchange, 26(2) (2000), 669-686.
  • [16] Kumar, V., On I and I*-convergence of double sequences, Math. Commun. 12 (2007), 171-181.
  • [17] Kumar, V. and Kumar, K., On the ideal convergence of sequences of fuzzy numbers, Inform. Sci. 178 (2008), 4670-4678.
  • [18] Matloka, M., Sequences of fuzzy numbers, Busefal, 28 (1986), 28-37.
  • [19] Mizumoto, M. and Tanaka, K., Some properties of fuzzy numbers, Advances in Fuzzy Set Theory and Applications, North-Holland (Amsterdam), 1979, 153-164.
  • [20] Mohiuddine, S.A., S. Şevli, H. and Cancan, M., Statistical convergence of double sequences in fuzzy normed spaces, Filomat, 26(4) (2012), 673-681.
  • [21] Mursaleen, M. and Edely, O.H.H., Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), 223-231.
  • [22] Nabiev, A., Pehlivan, S. and Gürdal, M., On I-Cauchy sequences, Taiwanese J. Math. 11(2) (2007) 569-5764.
  • [23] Nanda, S., On sequences of fuzzy numbers, Fuzzy Sets Syst. 33 (1989), 123-126.
  • [24] Pringsheim, A., Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), 289-321.
  • [25] Rath, D. and Tripaty, B.C., On statistically convergence and statistically Cauchy sequences, Indian J. Pure Appl. Math. 25(4) (1994), 381-386.
  • [26] Saadati, R., On the I-fuzzy topological spaces, Chaos, Solitons and Fractals, 37 (2008), 1419-1426.
  • [27] Salat, T., Tripaty, B.C. and Ziman, M., On I-convergence eld, Ital. J. Pure Appl. Math. 17 (2005), 45-54.
  • [28] Savaş, E. and Mursaleen, M., On statistically convergent double sequences of fuzzy numbers, Inform. Sci. 162 (2004), 183{-92
  • [29] S. Şençimen, C. and Pehlivan, S., Statistical convergence in fuzzy normed linear spaces,Fuzzy Sets and Systems, 159 (2008), 361-370.
  • [30] Schoenberg, I.J., The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361-375.
  • [31] Tripathy, B. and Tripathy, B.C., On I-convergent double sequences, Soochow J. Math. 31 (2005), 549-560.
  • [32] Turkmen, M. R. and Dündar, E., On Lacunary Statistical Convergence of Double Sequences and Some Properties in Fuzzy Normed Spaces, Journal of Intelligent and Fuzzy Systems, DOI: 10.3233/JIFS-18841 (Pre-press).
  • [33] Zadeh, L.A., Fuzzy sets, Information and Control 8(1965), 338-353.